Combinatorics

A 2nd-year Bachillerato student wants to set up a $4$-digit access code with different digits for their gym locker. They decide to use only the odd digits $\{1, 3, 5, 7, 9\}$ and cannot repeat any digit.

1. a) How many different codes can they create?

1. b) If the student forgets the code but remembers that the first digit was a $7$, how many different codes would they have to try at most?

In a robotics championship, a team must select $3$ representatives from a group of $8$ students to present the project to the panel of judges.

1. a) In how many different ways can the group of representatives be chosen if the selection order does not matter?

1. b) If among the $8$ students there are $2$ who specialise in programming and it is decided that at least one of them must be in the group, how many different selections are possible?

A streaming platform is designing a new custom interface. For the user profile, an avatar can be chosen by combining the following elements: 4 different face shapes, 5 types of hairstyle and 3 styles of glasses. In addition, the username must be a 5-character string consisting of 3 lower-case letters (from a 26-letter alphabet) followed by 2 digits.

1. a) How many different avatars can be created?
2. b) How many distinct usernames can be generated if neither letters nor digits can be repeated?
3. c) If a user decides to choose their avatar and username at random (under the conditions of part b), what is the probability that their name starts with the letter "z" and ends with the digit "9"?

Three phone operators, $M_1$, $M_2$ and $M_3$, handle $40\%$, $35\%$ and $25\%$ of customer service call centre calls, respectively. The percentage of calls that drop unexpectedly for each operator is $2\%$, $3\%$ and $4\%$, respectively.

A call is chosen at random and turns out to have dropped. Calculate the probability that it was handled by operator $M_1$.

A cybersecurity system uses two filters to detect phishing emails. Filter A detects $92\%$ of these emails, whilst Filter B detects $88\%$. Due to the structure of the system, an email is analysed by Filter B only if it has not been detected by Filter A. If an email is not detected by either filter, it reaches the user's inbox. It is estimated that $3\%$ of the emails received in the company are phishing attempts.

1. a) What is the probability that a phishing email reaches the inbox?
2. b) If an email has reached the inbox, what is the probability that it is actually a phishing email?

A cybersecurity team is analysing the vulnerability of an access system that uses passwords of $8$ characters. The allowed characters are the $26$ letters of the alphabet (case-insensitive) and the digits from $0$ to $9$. A password is considered "strong" if it meets the following conditions:

1. It contains exactly $2$ distinct numbers and $6$ letters.

2. The letters cannot be repeated.

3. The two numbers must be in consecutive positions.

Calculate:

1. a) The total number of strong passwords that can be generated.
2. b) If brute-force software generates a random password following only the structure of $8$ characters (letters or numbers), what is the probability that it guesses a specific strong password on the first attempt?

A courier company uses drones to deliver packages in a city. The probability of a drone suffering a technical breakdown during a delivery flight is $0,02$. The company makes $250$ daily deliveries independently.

1. a) Justify whether the variable "number of daily breakdowns" can be approximated by a normal distribution. If so, state its parameters.
2. b) Using the appropriate approximation, calculate the probability that on a given day exactly $5$ breakdowns occur.
3. c) Calculate the probability that between $3$ and $8$ breakdowns occur (inclusive).

A survey of 250 students at a technology centre was carried out regarding their use of operating systems. The results are shown in the following contingency table:

If a student is chosen at random, calculate:

1. a) The probability that they are a Higher Vocational student and mainly use Linux.
2. b) Given that the student mainly uses Windows, what is the probability that they are an Intermediate Vocational student?
3. c) Are the events "being a Higher Vocational student" and "mainly using Linux" independent? Give reasons for your answer.